What first conflict did miller see in the Salem trials that made him consider the play? What personal situation made him empathize with John Proctor?

Miller wrote The Crucible in direct response to Senator Joseph McCarthy's investigations (extremely controversial investigations) meant to root out Communist sympathizers here in the United States during the Cold War. These investigations stirred up a fervor much like those of the Salem witch trials (false allegations, ect.), and pitted one citizen against the others as amnesty was promised to those accused ready to confess or identify Red sympathizers.

As for the second part of your question, I think Miller answers it best.

"All this I understood. I had not approached the witchcraft out of nowhere, or from purely social and political considerations. My own marriage of twelve years was teetering and I knew more than I wished to know about where the blame lay. That John Proctor the sinner might overturn his paralyzing personal guilt and become the most forthright voice against the madness around him was a reassurance to me, and, I suppose, an inspiration: it demonstrated that a clear moral outcry could still spring even from an ambiguously unblemished soul. Moving crabwise across the profusion of evidence, I sensed that I had at last found something of myself in it, and a play began to accumulate around this man."